On Metric Structures of Normed Gyrogroups
© 2019, Springer Nature Switzerland AG. In this article, we indicate that the open unit ball in n-dimensional Euclidean space ℝn admits norm-like functions compatible with the Poincaré and Beltrami–Klein metrics. This leads to the notion of a normed gyrogroup, similar to that of a normed group in th...
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th-cmuir.6653943832-679122020-04-02T15:11:15Z On Metric Structures of Normed Gyrogroups Teerapong Suksumran Mathematics © 2019, Springer Nature Switzerland AG. In this article, we indicate that the open unit ball in n-dimensional Euclidean space ℝn admits norm-like functions compatible with the Poincaré and Beltrami–Klein metrics. This leads to the notion of a normed gyrogroup, similar to that of a normed group in the literature. We then examine topological and geometric structures of normed gyrogroups. In particular, we prove that the normed gyrogroups are homogeneous and form left invariant metric spaces and derive a version of the Mazur–Ulam theorem. We also give certain sufficient conditions, involving the right-gyrotranslation inequality and Klee’s condition, for a normed gyrogroup to be a topological gyrogroup. 2020-04-02T15:11:15Z 2020-04-02T15:11:15Z 2019-01-01 Book Series 19316836 19316828 2-s2.0-85076721163 10.1007/978-3-030-31339-5_20 https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85076721163&origin=inward http://cmuir.cmu.ac.th/jspui/handle/6653943832/67912 |
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Mathematics |
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Mathematics Teerapong Suksumran On Metric Structures of Normed Gyrogroups |
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© 2019, Springer Nature Switzerland AG. In this article, we indicate that the open unit ball in n-dimensional Euclidean space ℝn admits norm-like functions compatible with the Poincaré and Beltrami–Klein metrics. This leads to the notion of a normed gyrogroup, similar to that of a normed group in the literature. We then examine topological and geometric structures of normed gyrogroups. In particular, we prove that the normed gyrogroups are homogeneous and form left invariant metric spaces and derive a version of the Mazur–Ulam theorem. We also give certain sufficient conditions, involving the right-gyrotranslation inequality and Klee’s condition, for a normed gyrogroup to be a topological gyrogroup. |
| format |
Book Series |
| author |
Teerapong Suksumran |
| author_facet |
Teerapong Suksumran |
| author_sort |
Teerapong Suksumran |
| title |
On Metric Structures of Normed Gyrogroups |
| title_short |
On Metric Structures of Normed Gyrogroups |
| title_full |
On Metric Structures of Normed Gyrogroups |
| title_fullStr |
On Metric Structures of Normed Gyrogroups |
| title_full_unstemmed |
On Metric Structures of Normed Gyrogroups |
| title_sort |
on metric structures of normed gyrogroups |
| publishDate |
2020 |
| url |
https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85076721163&origin=inward http://cmuir.cmu.ac.th/jspui/handle/6653943832/67912 |
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